Properties

Label 1-1021-1021.144-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.368 + 0.929i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.572 + 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.412 − 0.911i)5-s + (−0.412 − 0.911i)6-s + (0.989 − 0.147i)7-s + (−0.966 + 0.255i)8-s + (0.932 + 0.361i)9-s + (0.510 − 0.859i)10-s + (−0.659 + 0.751i)11-s + (0.510 − 0.859i)12-s + (−0.602 − 0.798i)13-s + (0.687 + 0.726i)14-s + (0.237 + 0.971i)15-s + (−0.763 − 0.645i)16-s + (−0.993 + 0.110i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.819i)2-s + (−0.982 − 0.183i)3-s + (−0.343 + 0.938i)4-s + (−0.412 − 0.911i)5-s + (−0.412 − 0.911i)6-s + (0.989 − 0.147i)7-s + (−0.966 + 0.255i)8-s + (0.932 + 0.361i)9-s + (0.510 − 0.859i)10-s + (−0.659 + 0.751i)11-s + (0.510 − 0.859i)12-s + (−0.602 − 0.798i)13-s + (0.687 + 0.726i)14-s + (0.237 + 0.971i)15-s + (−0.763 − 0.645i)16-s + (−0.993 + 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5603463833 + 0.8244689475i\)
\(L(\frac12)\) \(\approx\) \(0.5603463833 + 0.8244689475i\)
\(L(1)\) \(\approx\) \(0.8299961997 + 0.3713762564i\)
\(L(1)\) \(\approx\) \(0.8299961997 + 0.3713762564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.572 + 0.819i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.412 - 0.911i)T \)
7 \( 1 + (0.989 - 0.147i)T \)
11 \( 1 + (-0.659 + 0.751i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (-0.993 + 0.110i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (-0.763 + 0.645i)T \)
29 \( 1 + (0.989 + 0.147i)T \)
31 \( 1 + (-0.0554 + 0.998i)T \)
37 \( 1 + (0.903 - 0.429i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (-0.713 + 0.700i)T \)
47 \( 1 + (0.687 + 0.726i)T \)
53 \( 1 + (0.378 + 0.925i)T \)
59 \( 1 + (-0.128 - 0.991i)T \)
61 \( 1 + (0.687 + 0.726i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.966 + 0.255i)T \)
73 \( 1 + (0.687 - 0.726i)T \)
79 \( 1 + (0.975 + 0.219i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.61048347635173744287053828191, −20.814098093988318105310185918378, −19.88178028244954592976653144907, −18.887287040154298056181194902350, −18.261040835969673623926416267968, −17.88380564116632198228534872375, −16.602497958077786984392937210614, −15.682210738754132881625780417075, −15.00687877484254219895241841683, −14.15613763766311397744228163339, −13.47113125834444876727310549610, −12.242742799422333158359263874684, −11.59207304702341943754584136734, −11.230273837117286097565236302435, −10.42826836758184926274012150028, −9.73604259080096603535465368718, −8.439051150740983481944241672491, −7.31524523046665667226679724687, −6.34085649244907358740691832835, −5.58070173438965850028560079699, −4.605680810974098779305181888894, −4.058482155717208499835606142375, −2.77526181371013075670056691710, −1.91926070170243481761837829667, −0.47323592734217211857637474670, 1.0270645367275458085563896947, 2.43356810639670019475679939241, 4.03977487434206909829655698179, 4.86157625325315289223908535084, 5.107288750041927823556911542657, 6.082749033812012741361582631264, 7.45804231084403791711065426588, 7.568568049755735562100147770436, 8.619057419998585706687391581840, 9.74616058103131823517254878242, 10.89016764636809594429870920601, 11.78891894661306488608654230974, 12.36888351576651636983847674870, 13.04684001343124030764329744506, 13.84273690694965896031466624414, 14.995839716950296306104143762027, 15.73024401627084849853261431571, 16.161955482530341010714709574616, 17.24860845546014312551325074583, 17.81302390621732316105254280218, 18.02475680570455606845802960470, 19.70973665147754368485578750475, 20.423294622238969975685158121638, 21.30706180492465381504588840215, 21.98257467822930271114894254251

Graph of the $Z$-function along the critical line